Differentiability and Density Continuity


Krzysztof Ciesielski & Lee Larson & Krzysztof Ostaszewski

Real Anal. Exchange 15 (1989-90), 239-247

The density topology on R consists of all measurable subsets A of R such that, for every x in A, x is a density point of A. It is a completely regular refinement of the natural topology. A function f:R-->R is density continuous if and only if it is continuous as a selfmap of R equipped with the density topology.

Throughout this paper we are concerned with the relationship between density continuity and differentiability. In the process, we discuss the fact that any closed set can be made into the zero set of a Cinfty density continuous function, and we show that there is a nowhere approximately differentiable density continuous and continuous function. This example answers a problem posed by Ostaszewski.

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Last modified August 23, 1999.